Because it is known to be true that a correct pure simulation of the
input by the halt decider would demonstrate the actual behavior of this
input it is known that this is a correct basis for a halt status
decision by the halt decider.

A simulating halt decider need only perform a pure simulation of N steps
of its input until:
(a) Its input halts on its own or
(b) The simulated behavior of its input correctly matches non-halting
behavior patterns conclusively proving that a pure simulation of this
input would never stop running.

[*Olcott 2021 generic halt deciding principle*] Whenever the pure
simulation of the input to simulating halt decider H(x,y) never stops
running unless H aborts its simulation H correctly aborts this
simulation and returns 0 for not halting.

*Computable functions* are the basic objects of study in computability
theory. Computable functions are the formalized analogue of the
intuitive notion of algorithms, in the sense that a function is
computable if there exists an algorithm that can do the job of the
function, i.e. given *an input of the function domain it can return the
corresponding output* https://en.wikipedia.org/wiki/Computable_function

A halt decider is a decider; which is a type of computable function;
which is a type of function; which only applies to its inputs in its domain.

No function is ever accountable for anything besides input parameters in
its domain.

This means that when int main() { P(P); }; has different behavior than
int main() { H(P,P); }; this does not make any difference at all.

*Halting problem undecidability and infinitely nested simulation V2*
https://www.researchgate.net/publication/356105750_Halting_problem_undecidability_and_infinitely_nested_simulation_V2

--
Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer